Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications

Abstract

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$-\Psi (x,|\nabla u|){\Delta }_p^{\mathrm{N}}u(x)+\mathcal{H}(x,\nabla u)=f(x)\mathrm{in}\Omega ,\mathrm{for}1<p<\infty ,$$ under suitable assumptions on the data of the problem. Particularly, our results are relevant for a class of quasi-linear models with Hamiltonian terms. Additionally, we address non-degeneracy estimates for such solutions and present a couple of applications.

arXiv comment28 pages